Formal Naive Dirac Operators and Graph Topology
G. M. von Hippel
TL;DR
The paper tackles the Misumi–Yumoto conjecture by building a graph-based formal Dirac operator framework on commutative, fully-even $d$-Dirac graphs and showing that zero modes bound not only the total but also individual Betti numbers $b_r$ via a $W_G$ doubler-count matrix. It develops a Künneth-type decomposition for box products, connects zero modes to quotient graphs $G^ lat$ and to abelian-group representations, and constructs a cubical homology on $G$ that yields exact Betti-number bounds $m_r(G)\le b_r(G)$, with $b_r(G)=inom{d}{r}$ in torus-like cases. The results tie the topology of the underlying graph to the spectrum of the Dirac operator, providing a concrete, graph-theoretic realization of the topology-Dirac operator correspondence and offering a path toward analyzing more general noncommutative or non-fully-even cases. This work thus deepens the link between lattice fermion doublers and topological invariants in a discrete setting, with potential implications for lattice gauge theories and discrete topology. All key relations are expressed with formal Dirac operators $D_G$, graph Laplacians $ riangle_G$, and Betti numbers $b_r(G)$, highlighting how group representations govern zero modes on abelian Cayley-graphs.
Abstract
Motivated by a recent conjecture of Misumi and Yumoto relating the number of zero modes of lattice Dirac operators to the sum of the Betti numbers of the underlying spacetime manifold, we study formal Dirac operators on a class of graphs admitting such in terms of their zero modes. Our main result is that for graphs on which translations commute, the conjecture of Misumi and Yumoto can be shown and indeed can be strengthened to obtain bounds on the individual Betti numbers rather than merely on their sum. Interpretations of the zero modes in terms of graph quotients and of the representation theory of abelian groups are given, and connections with a homology theory for such graphs are highlighted.